This invention relates to methods and apparatuses for carrying out optical time domain reflectometry, particularly to the use of weighing techniques to improve the signal-to-noise ratio of reflected optical power measurements in optical time domain reflectometry.
In optical time domain reflectometry, as applied to a fiber optic cable, a pulse of light is launched, or transmitted, down the fiber from one end and the amount of light power reflected back to that end is measured as a function of time. Since the velocity of light in the fiber is known, the power reflected from any given distance down the fiber can therefore be determined. The operations of launching a pulse into the fiber and detecting the optical power level leaving the fiber are repeated for a large number of different time delay values, and the resulting data is used to form a display that represents fiber condition as a function of distance. This allows the quality of the fiber to be determined and, often more importantly, permits the existence of any significant discontinuity in the fiber or fiber system to be identified and its effect to be measured.
The reflected power comes from either Fresnel reflections, off a connector for example, or from Rayleigh backscatter. Rayleigh backscatter results from randomly dispersed anomolies in the index of refraction of the fiber. In contrast to the Fresnel reflections, Rayleigh backscatter is very weak, typically 40 dB or more less than the launch power. Consequently, the ability of an optical time domain reflectometer ("OTDR") to measure Rayleigh backscatter is limited by the noise in the receiver itself, and various techniques have been used to increase the signal-to-noise ratio.
In addition, the distance resolution of an OTDR customarily depends on the length of the pulse in the fiber (pulse period times the velocity of the light in the fiber). More specifically, the resolution is one half the pulse length. However, the amplitude resolution is directly proportional to the total energy of the pulse (pulse period times the power of the energy source). Since the light source has a fixed maximum power, there is a tradeoff between distance resolution and amplitude resolution. This tradeoff can be alleviated by increasing the signal-to-noise ratio of the reflected energy, thereby increasing the amplitude resolution or, alternatively, permitting the distance resolution to be increased by decreasing the pulse length.
Techniques used to increase the signal-to-noise ratio include high speed averaging, spread spectrum techniques using Golay codes and coherent optical detection. Correlation techniques using various codes are described in P. Healy, "Complementary Correlation OTDR With Three Codewords", Electron. Lett., 1990, 26, pp. 70-71; P. Healy, "Complementary Code Sets for OTDR", Electron. Lett., 1989, 25, pp. 692-693; P. Healy, "Optical Orthogonal Pulse Compression Codes by Hopping", Electron. Lett., 1981, 17, pp. 970-971. The use of Golay codes is described in Cheng et al. U.S. Pat. No. 4,743,753. While these techniques have been effective, further increases in sensitivity and signal-to-noise ratio are still needed.
In the invention described hereafter weighing designs have been employed to improve the signal-to-noise ratio of an optical time domain reflectometer. A weighing design is a scheme for accurately weighing a number of objects by weighing them in groups rather than one at a time. Where the error in weighing is independent of the weight of the objects, the mean square error .epsilon. can be reduced by a weighing design. Where i objects are weighed separately, the measured weight .mu..sub.i of the ith object is the actual weight .tau..sub.i plus some error e.sub.i. That is: EQU .mu..sub.i =.tau..sub.i +e.sub.i
The best estimate of the weight .tau..sub.i can therefore be expressed as .tau..sub.i +e.sub.i. It follows that: EQU e.sub.i =.tau..sub.i -.tau..sub.i
The expected value of the square of the error, that is, the mean square error, is expressed as E{e.sub.i.sup.2 }, where E denotes expected value. For a single measurement the mean square error is equal to the variance .sigma..sup.2. Here, it can also be expressed as: EQU E{(.tau..sub.i -.tau..sub.i).sup.2 }=.upsilon..sup.2
For example, suppose that the weighing is performed using a chemical balance with two pans, and that four objects are to be weighed, as follows: EQU .mu..sub.1 =.tau..sub.1 +.tau..sub.2 +.tau..sub.3 +.tau..sub.4 +e.sub.1 EQU .mu..sub.2 =.tau..sub.1 -.tau..sub.2 +.tau..sub.3 -.tau..sub.4 +e.sub.2 EQU .mu..sub.3 =.tau..sub.1 +.tau..sub.2 -.tau..sub.3 -.tau..sub.4 +e.sub.3 EQU .mu..sub.4 =.tau..sub.1 -.tau..sub.2 -.tau..sub.3 +.tau..sub.4 +e.sub.4
This means that in the first weighing all four objects are placed in the left hand pan, in the second weighing objects 1 and 3 are placed in the left hand pan, and 2 and 4 in the right hand pan, and so forth.
The best estimate of the weights can be found by solving these equations simultaneously for .tau..sub.1, .tau..sub.2, .tau..sub.3 and .tau..sub.4. For example, it can be shown that: EQU .tau..sub.1 =1/4(.mu..sub.1 +.mu..sub.2 +.mu..sub.3 +.mu..sub.4)
and that: EQU .tau..sub.1 =.tau..sub.1 +1/4(e.sub.1 +e.sub.2 +e.sub.3 +e.sub.4)
Consequently, the mean square error is: ##EQU1## In general, it can be shown for this case that: EQU E{(.tau..sub.i -.tau..sub.i).sup.2 }=1/4.sigma..sup.2
That is, by weighing the four objects together, the mean square error can be reduced by a factor of 1/4.
The signal-to-noise ratio for a weighing design is defined as the ratio of the true weight of the ith object divided by the standard deviation .sigma. of the error in the measurement. That is: ##EQU2##
Another weighing design is based on a spring balance having only one pan. Only coefficients 0 and 1 can be used; that is, an object is either weighed or it is not. For example, one method of weighing is: ##EQU3## By solving these equations simultaneously for .tau..sub.1 through .tau..sub.3, it can be shown that the mean square errors are reduced. For example: ##EQU4##
Weighing designs are described by a square, N.times.N matrix, where the rows of the matrix correspond to the weighing event and the columns correspond to the position of the object in the weighing event. The matrix therefore acts as a transform that changes a set of measurements to a set of equations. In a chemical balance weighing design the values in the matrix may take on +1, 0 or -1. In a spring balance weighing design the values may only be 0 or +1.
The best weighing designs use Hadamard matrices for chemical balance designs and simplex matrices ("S-matrices") for spring balance designs. A Hadamard matrix H.sub.N of order N is an N.times.N matrix of +1's and -1's with the property that the scalar product of any two distinct rows is 0. A normalized Hadamard matrix is one in which all elements of the first row and the first column are +1. If H.sub.N is a normalized Hadamard matrix, an S-matrix is the (N-1).times.(N-1) matrix of 0's and +1's obtained by (a) omitting the first row and the first column of H.sub.N, and (b) changing the +1's to 0 and the -1's to 1's. The rows of an S-matrix are codewords in a simplex code. The rows of a Hadamard matrix are referred to herein as codewords in a Hadamard code and are orthogonal.
Ideally, a weighing design should minimize the mean square error .epsilon..sub.i for all i simultaneously. Where this is impossible, other criteria are used. A weighing design is said to be "A-optimal" if it minimizes the average of the mean square error, that is: ##EQU5##
It can be shown that the average mean square error for a Hadamard matrix is: ##EQU6## and that the average mean square error for an S-matrix is given by: ##EQU7## For large N, S-matrices are within a few percent of being A-optimal.
The forgoing information has been derived from M. Harwit and N. Sloan, Hadamard Transform Optics., 6-17 (1979, Academic Press, Inc.), ISBN 0-12-330050-9.